Abstract
Numerical modeling is a promising way to understand the characteristics of the thermo-mechanical (TM) coupled behaviors in rocks under high-temperature impact. This paper documents the thermo-mechanical response of Eibenstock granite (EG) through laboratory experiments and numerical simulations using a proposed TM coupled Grain-Based Model (GBM). Uniaxial compression and Brazilian tests of EG specimens after 400 °C and 600 °C heating–cooling cycles were undertaken. Based on the laboratory results, a newly developed TM coupled contact constitutive law considers mineral composition, heterogeneous temperature-dependent properties, reversible α ↔ β quartz-transitions, crack-slipping displacements with strength reduction, and real-time crack evolution. The models can well reproduce the real-time thermal expansion–contraction, microstructural changes, nonlinear stress–strain behavior, temperature-dependent strength, and the ultimate failure modes of thermal-damaged specimens. The P-wave velocities and the simulated thermal cracking revealed that the material contraction during cooling leads to a width reduction of the earlier heating-formed cracks. Newly induced microcracks are rare during cooling due to the released stress concentrations of the local mineral grains. The residual thermal strain, which results from microcrack formation and the growth of pre-existing microcracks, was simulated and used as a quantitative index of thermally induced damages. The damage degree of the 600 °C samples was up to six times higher than that of the 400 °C, leading to a stronger strength reduction upon mechanical loading and a higher concave stress–strain nonlinearity caused by micro-crack closing at the beginning of loading. In general, the GBM is able to simulate the TM coupled behavior of polycrystalline rocks in a realistic manner.
Highlights
-
A joint constitutive law considering both temperature and crack-slipping is proposed.
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TM models considering a heating-cooling cycle are calibrated by laboratory tests.
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Real-time thermal-induced strains of granite specimens are reproduced.
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Concave stress-strain nonlinearity of thermally damaged granites is replicated.
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Abbreviations
- A c :
-
Area of the contact
- B :
-
Thickness of the Brazilian disc
- C p :
-
Specific heat
- D :
-
Diameter of the Brazilian disc
- E m :
-
Micro-Young’s modulus
- F max :
-
Maximum load
- F s :
-
Shear force magnitude
- F i s :
-
Shear force vector
- F n :
-
Normal force
- K :
-
Bulk modulus
- M :
-
Mass
- Q i :
-
Heat flux in the positive i-direction
- Q net :
-
Net heat flow
- S max :
-
Maximum shear force
- S res :
-
Residual shear force
- S temp-max :
-
Maximum shear force depending on temperature
- \(S_{{{\text{temp}} - {\text{res}}}}^{{{\text{disp}}}}\) :
-
Residual maximum shear force considering contact slipping displacement and temperature
- T max :
-
Maximum tensile normal force
- T res :
-
Residual tensile normal force
- T temp-max :
-
Maximum tensile normal force dependent on temperature
- c m :
-
Micro-cohesion
- c 0 :
-
Initial cohesion before heat treatment
- c temp :
-
Temperature-dependent cohesion
- c res :
-
Residual cohesion
- \(c_{{{\text{res}}}}^{{{\text{disp}}}}\) :
-
Residual cohesion depending on contact slipping displacement
- \(c_{{{\text{temp}} - {\text{res}}}}^{{{\text{disp}}}}\) :
-
Residual cohesion depending on contact slipping displacement and temperature
- \(f_{{{\text{res}} - c}}^{{{\text{disp}}}}\) :
-
Slip-weakening function of residual cohesion depending on contact slipping displacement
- \(f_{{{\text{res}} - \varphi }}^{{{\text{disp}}}}\) :
-
Slip-weakening function of residual friction angle depending on contact slipping displacement
- f temp -c :
-
Temperature-strengthening functions of cohesion
- f temp -φ :
-
Temperature-strengthening functions of friction angle
- f temp -t :
-
Temperature-strengthening functions of tensile strength
- h sample :
-
Height of the sample
- k :
-
Thermal conductivity
- k n :
-
Contact normal stiffness
- k s :
-
Contact shear stiffness
- k ij :
-
Thermal conductivity tensor
- t step :
-
Mechanical timestep
- t model :
-
Model time
- v y :
-
Mechanical loading velocity
- v step :
-
Steps per second
- α t :
-
Coefficient of linear thermal expansion for specific temperature
- ε c :
-
Failure strain
- δ ij :
-
Kronecker delta
- σ t :
-
Tensile stress
- σ temp -t :
-
Temperature-dependent tensile strength
- ρ m :
-
Micro-density
- ρ :
-
Mass density
- ν m :
-
Micro-Poisson’s ratio
- φ m :
-
Micro-friction angle
- φ temp :
-
Temperature-dependent friction angle
- \(\varphi_{{{\text{res}}}}^{{{\text{disp}}}}\) :
-
Residual friction angle depending on contact slipping displacement
- \(\varphi_{{{\text{temp}} - {\text{res}}}}^{{{\text{disp}}}}\) :
-
Residual friction angle depending on contact slipping displacement and temperature
- ∆F n :
-
Normal force increment
- ∆F i s :
-
Shear force vector increment
- ∆h:
-
Height increment
- ∆u n :
-
Normal displacement increment
- ∆u i s :
-
Shear displacement vector increment
- ΔT :
-
Temperature increment
- ∆z min :
-
Smallest width of an adjoining zone
- ∆σ ij :
-
Stress increment
- ∆ε ij :
-
Strain increment
- ∂T/∂x j :
-
Temperature gradient
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (No. 52104120). The cooperation with Dr. Thomas Frühwirt and Mr. Weinhold, who were involved in lab testing in the rockmechanical laboratory at TU Bergakademie Freiberg, is highly acknowledged.
Funding
This work was supported by the National Natural Science Foundation of China (No. 52104120).
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Conceptualization: FW; methodology: FW; formal analysis and investigation: FW, HK, MI; writing—original draft preparation: FW; writing—review and editing: HK, MI, BP, RP, YZ; funding acquisition: FW; resources: FW, RP, YZ.
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Wang, F., Konietzky, H., Pang, R. et al. Grain-Based Discrete Element Modeling of Thermo-Mechanical Response of Granite under Temperature. Rock Mech Rock Eng 56, 5009–5027 (2023). https://doi.org/10.1007/s00603-023-03316-0
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DOI: https://doi.org/10.1007/s00603-023-03316-0